Assgmet 5/MATH 7/Wter 00 Due: Frday, February 9 class (!) (aswers wll be posted rght after class) As usual, there are peces of text, before the questos [], [], themselves. Recall: For the quadratc form x qx ( ) = qx (, x,..., x) = aj x xj ( X= ),, j= x we have the max-form qx ( ) = X AX ( A= ( a j ) ). y Moreover, f s a vertble max, ad X = Y ( Y = ), the y qx ( ) = qy ˆ( ) = qy ˆ(, y,..., y) = Y BY for the max B = A. We assume that ( ) A= a j s symmec: aj = aj for all, j =,..., ; as a cosequece, the B s also symmec. The quadratc form qx ( ) s ) postve defte, ) egatve defte, 3) defte f, respectvely: ) for all X 0, we have qx ( ) > 0 ; ) for all X 0, we have qx ( ) < 0; 3) there are X 0, X 0 such that qx ( ) > 0 ad qx ( ) < 0. These three qualtes are mutually exclusve; but a quadratc form may be oe of the above three kds. qx ( ) s ) postve semdefte, 5) egatve semdefte f, respectvely: ) for all X 0, we have qx ( ) 0; 5) for all X 0, we have qx ( ) 0. Our text says oegatve semdefte for postve semdefte. The termology s such that postve semdefte s more geeral tha postve defte ; ad smlarly for egatve semdefte. However, the mportat possblty s that qx ( ) may be postve semdefte, wthout beg postve defte. For stace,
qxy (, ) x xy y = + + s postve semdefte sce qxy (, ) = ( x+ y) 0 always; but 0 ad q( ) = q(, ) = ( ) = 0, thus q s ot postve defte. Of course, f defte. q s postve semdefte, the t caot be ether egatve defte, or (I the ma applcato aalyss (calculus), whe the quadratc form formed by the secod partals at a statoary pot s postve defte, oe has a local mmum; whe the quadratc form s egatve defte, oe has a local maxmum; whe t s defte, oe deftely has o local mmum, or a maxmum; ad whe t s semdefte but ot defte, oe does ot kow about local exemum wthout further vestgato. Ths wll be looked at later class.) Wth the quadratc form qx ( ) = X AX, we assocate the symmec blear form ( X, X ) defed by β ( X, X ) X A X β = β ( X, X ) = β ( X, X ) We have the detty qx) ( = β ( XX, ).. β s symmec the sese that [] Let qx ( ) = qx (, yzw,, ) = x y 3z + 5w+ 5xz+ xw+ wz (X Throughout ths questo, q deotes ths partcular quadratc form. x y = ). z w ) Wrte qx ( ) the symmec max form. r s ) Apply the coordate asformato X = Y, Y =, wth the t u asformg max = 0 5/ 7/37 0 0 0 0 0 8/37 0 0 0
ad wrte qx ( ) the form qxyzw (,,, ) = qrstu ˆ(,,, ) = byy j j (where we wrote y = r, y = s, y = t, y = u) wth sutable coeffcets. Use precse fracto 3 calculato, rather tha decmals; the result should come out precsely dagoal. b j, j= 3) Usg the max approprately, wrte each of the varables rstu,,, as a lear expresso of x, yzw,, ; ad verfy drectly the resultg detty qxyzw (,,, ) = qrs ˆ(,, tu, ). ) Usg the work doe so far, decde what kd q s the defteess classfcato (see above!). rove your asserto: show that q satsfes the defto gve above of the qualty (postve defte,, defte, ). Appealg to some theorem about that qualty s ot eough. 5) Cosder the symmec blear form β assocated wth q. Let X = 0, Y =. 0 3 Calculate β ( X, X)( = q( X)), β ( YY, ) ad β ( X, Y ), ad gve a formula for qax ( + by) the form of a homogeeous quadratc polyomal (quadratc form!) of the varables a ad b. The Gram-Schmdt orthogoalzato process Read secto 7.7, p. 6, the text. Whe you see er product space V, take V to be ay subspace of a stadard space, ad for the er product uv,, read dot product: X, Y = X Y. Note Remark o page 6 especally. Read Example 7.0 o the same page, but omt the ext example 7. (whch s a vector space of a kd that we have ot looked at yet). May also read solved problem 7., p.6. You wll eed to use Gram-Schmdt some problems that follow. 3
Recall: A max s called orthogoal f = = I ; equvaletly, f ; ad equvaletly, f the colums,..., of form a orthoormal system: = 0 whe j j; ad = =. If U = ( u,..., u ) ad V = ( v,..., v ) both m are orthoormal bases of a subspace V of some ( partcular whe V =, U = E, the stadard bass, ad V s aother orthoormal bass of ), the = [ U V ], the chage-of-bass max, s a orthogoal max. [] ) rove that f ad Q are orthogoal maces, the so are,, Q. ) Use Gram-Schmdt to fd a orthogoal max the frst colum of / / whch s =. / / Isuctos: Start wth ay bass B of the frst of whose vectors s. You may use some of the stadard vectors e,..., e as the last three vectors of B. Apply Gram- Schmdt to B. Clear deomators termedate results too. Use the symbol to / 3/ 3 3 deote a scaled vector: e.g., =. Scalg should be used also to 3/ 6 6 / 3 6 get smaller umbers: e.g.,. Normalze the system at the ed oly. Wrte 9 3 precse fractos ad square-roots whe ecessary; do ot use decmal approxmato. 3 3) Wth determed ), ad deotg the colums of by,,,, let U = spa(, ). Use ) (that s: do t start from scratch!) ad gve a bass of U, the orthogoal complemet of U.
7 ) For U as 3), ad Y =, determe proj ( 0 Y U ) ad proj U ( Y) (Ht: read Theorem 7.8, page 5 ( just before the secto o Gram-Schmdt), ad the Remark followg t. Take ote of the term Fourer coeffcet just before the remark. It wll come back later.) [3] We cosder the followg quadratc forms: q ( X ) = x + 3y z w + xy+ xz+ xw yz yw+ 6zw, q ( X ) = x + y + xy+ xz+ xw+ zw q ( X ) = 3 x 3z 3w yz yw + zw q ( X ) = 3x + 5y + 5z + 5w yz yw zw Treat each of the quadratc forms q = q, q,... as follows. ) Fd a orthogoal asformg max that turs qx ( ) to a puresquare (dagoal) form qy ˆ( ) terms of the ew varables Y ; wrte out what qy ˆ( ) s, ad what the relatoshp s betwee the old ad ew coordate vectors X ad Y. ) Determe whch kd q s the defteess classfcato. For each case whe the q s defte, or semdefte-o-defte, supply examples of vectors X that show the property questo. ( ) 3) Determe max qx X ( ) ad m qx X. ) Suppose that A s a symmec max, s a orthogoal max, ad A s a dagoal. Let B be a polyomal expresso of A : k B = bi 0 +ba +... + ba k. rove that B s symmec, ad B s dagoal. (Hts: f you fd t too hard, verfy the asserto frst for the specal cases B = A, A, I + A, I + 3A+ A,... utl the patter becomes clear.) 5
5) Cosder the quadratc form q ( X ) = 5 x + y + 3z + 3w + xy+ xz+ xw+ yz+ yw+ zw. Verfy that ( ) q5 X = X B X for the max B = A + A+ I, where A s the max for whch ( ) q X = X A X, wth q from above. Usg kowledge o q ( X), do the work asked for ), ) ad 3) for q ( X), wth a mmal amout of calculato oly. 5 6) Let A be a symmec max, qx ( ) = X AX the correspodg quadratc form. rove that the fucto QX ( ) defed by QX ( ) = qax ( X) s a quadratc form, ad ay orthogoal max that asforms q to pure-squares form, does the same for Q. Determe the values from the values λ for whch qx ( ) = λ y. = μ for whch qx ( ) = μ y ( X = Y ) = 6